• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1211-1217
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• 2019

Let $0<{\alpha}<{\infty}$ be fixed, and let $X=(X_t)_{t{\geq}0}$ be a Bessel process with dimension $0<{\theta}{\leq}1$ starting at $x{\geq}0$. In this paper, it is proved that there are positive constants A and D depending only on ${\theta}$ and ${\alpha}$ such that $$E_x\({\exp}[{\alpha}\;\max_{0{\leq}t{\leq}{\tau}}\;X_t]\){\leq}AE_x\({\exp}[D_{\tau}]\)$$ for any stopping time ${\tau}$ of X. This inequality is also shown to be sharp.

• Bulletin of the Korean Mathematical Society
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• v.20 no.2
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• pp.99-103
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• 1983

Internal heat generation is one of the insidious conditions affecting the quality of an industrial product after it is cast, coated, molded, forged or laminated. Frequently, the product is pressed into service before the exothermic chemical reactions in the generic material has been completed. The heat liberated from this continuing chemical reaction or the residual deformation from the rheological activities in the materials must be adequately removed or prevented, or the product may be discolored, warped, weakened or even "ignited" spontaneously. Numerous instances of premature structural failures, product-recalls, and/or system-malfunctions have been recorded in recent history. The Coulee Dam was poured with pre-chilled concrete just to negate this freakish encore. It is well-known that concrete (a non-isotropic conducting medium), for instance, takes 28 days to develop its full strength. During this period of curing it is conceivable that the processes of internal heat generation, heat conduction and heat dissipation take place simultaneously inside the medium.he medium.